September 28, 2017 Thursday
Bedtime Story
The Syntax and Essence of Lambda Calculus
Now having this much knowledge, let us go
directly into the syntax of lambda calculus.
Please do not get scared by the word
syntax.
Syntax simply means notations and the set
of rules that are made which will govern a mathematical system or a language.
After all, even sports are governed by
rules and regulations.
Using this syntax devised by Church, you
can define a function which after all was the whole point.
The first symbol that comes in the syntax
of lambda calculus is of the small case Greek letter lambda or λ.
So how can you write down the function that
we took as our first example where any input x would come as x + 10.
This is how.
λx.x+10
This is read out as lambda x dot x plus
ten.
Doesn’t seem to very difficult, does it?
Now I am sure you can easily figure out how
to write down the second function that we considered in the syntax of lambda
calculus.
λx.λy.x+y
This is read out as lambda x dot lambda y
dot x plus y.
So what can we do with a function so
created with this weird lambda notation?
We can perform operation on specific
numbers which are substituted into the variables like x and y.
So if we take the first incremental
function and feed in the value of 10 in the black box, we get the output of 20.
This is in essence is the gist of lambda
calculus.
So lambda calculus comprises of three
things:
(1) Variables such as x or y
(2) A way of building function using λ
notation
(3) A way of applying function
When you see how simple it is, you must be
wondering what is so great about it and why to make a big fuss of it.
What was the whole point in simply devising
a new notion to the idea of function?
To begin with, Church was not motivated by
the idea of computing, programming or even computable functions.
Being a student of David Hilbert at Gottingen,
he was purely interested in attacking the foundations of mathematics as the existing
system of Curry’s combinatory logic which we will come to later was shown to
have inconsistencies.
This was demonstrated by Stephen Kleene and
J. B Rosser (both students of Alonzo Church) through now what is famously known
as the Kleene-Rosser paradox.
Stay tuned to the voice of an average story storytelling
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Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
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recommend this large collection of Halloween Songs for Kids:
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